text: A mathematical description of vacuum Rabi oscillation begins with the Jaynes–Cummings model, which describes the interaction between a single mode of a quantized field and a two level system inside an optical cavity. The Hamiltonian for this model in the rotating wave approximation is
  where  is the Pauli z spin operator for the two eigenstates  and  of the isolated two level system separated in energy by ;  and  are the raising and lowering operators of the two level system;  and  are the creation and annihilation operators for photons of energy  in the cavity mode; and
  is the strength of the coupling between the dipole moment  of the two level system and the cavity mode with volume  and electric field polarized along .
  The energy eigenvalues and eigenstates for this model are 
  where  is the detuning, and the angle  is defined as
  Given the eigenstates of the system, the time evolution operator can be written down in the form
  If the system starts in the state , where the atom is in the ground state of the two level system and there are  photons in the cavity mode, the application of the time evolution operator yields
  The probability that the two level system is in the excited state  as a function of time  is then
  where  is identified as the Rabi frequency. For the case that there is no electric field in the cavity, that is, the photon number  is zero, the Rabi frequency becomes . Then, the probability that the two level system goes from its ground state to its excited state as a function of time  is
  For a cavity that admits a single mode perfectly resonant with the energy difference between the two energy levels, the detuning  vanishes, and  becomes a squared sinusoid with unit amplitude and period